Problem: All of the 4th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$2.50$ each for students, and the group paid $$32.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$11.00$ each for students, and the group paid $$134.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2.5y = 32}$ ${34x+11y = 134}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-10y = -128}$ ${34x+11y = 134}$ Add the top and bottom equations together. ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+2.5y = 32}$ to find $x$ ${8.5x + 2.5}{(6)}{= 32}$ $8.5x+15 = 32$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {34x+11y = 134}$ and get the same answer for $x$ ${34x + 11}{(6)}{= 134}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.